An Existence Theorem for the Reduced Wave Equation

Abstract

We consider here the two dimensional problem of scattering of a wave by a finite set of smooth, finite, nonintersecting arcs. Let the points in E2 he denoted by z = x+iy. Let £,-, i -l, • • • , n he arcs given by z = Xi(t) + iyi(t), 0 á i á 1, i = 1, ».We denote the point x¿(0)+íy,(0) by a¿ and the point Xi(l)+iy((l) by bi. We assume that the functions x¿(í) and y((t) have Holder continuous second derivatives and that the arcs £,-do not intersect. We denote the union of the £,'s by £ and the open set £2 -£ by G. We seek a function u,(x, y) which satisfies the following conditions: Uniqueness follows from the work of Levine [2]. (Levine proves his uniqueness theorem in the three dimensional case, however his proof can easily be modified so as to apply here.) Here we will prove existence. Our method is similar to that of Leis [l] who considered the case of scattering by a piecewise smooth closed contour.

On the Inverse of an Integral Operator

“…Then given/(x) such that/' is Holder continuous, equation (1) has a unique solution, <pEh. In this paper we will consider equation (1) as a mapping from one Hilbert space into another.…”

Numerical Solution of the Scattering Problem for Acoustic Waves by a Two-Sided Crack in 2-Dimensional Space

“…Hence, compared with the scattering problems of obstacles with closed smooth boundary, these crack scattering problems are more complicated due to the eventual different surface impedances on both sides as well as the presence of the tips of crack. The inverse scattering problems for arc with Dirichlet or Neumann data on the arc has been considered in [12,14,17,18,22,30,33,34] using iterative methods. Physically, this means that the arc is sound-soft or sound hard.…”

The oblique derivative problem for the Helmholtz equation and scattering tidal waves

“…Scattering of acoustic, electromagnetic and elastic waves by screens, cracks or wings often lead to solving boundary-value problems in the exterior of open surfaces (Davydov et al 1981;Durand 1983;Grisvard 1985;Krutitskii 1994aKrutitskii , b, 1996aKrutitskii -c, 1998Krutitskii , 2000Lifanov 1996;Muskhelishvili 1972;Povzner & Suharevsky 1959;Proudman 1953;Tuchkin 1987;Tuchkin & Shestopalov 1985;Wolfe 1969;Zakharov & Sobyanina 1986). If screens take the form of half-planes, the problem can sometimes be solved explicitly by the Wiener-Hopf technique (Crease 1956(Crease , 1958Noble 1958;Santos & Speck 1996).…”